# Value of a European Call option with Infinite maturity

It is a job interview question.

So, what's the value of a vanilla European call option of infinite maturity, and a given strike, vol, interest rate, spot price.

I think, the answer should be "zero".

The contract never pays, because infinite maturity will never be reached.

It should not be equal to the spot price, which BS formula suggests in the limit T goes to infinity, I think.

If it were an American call with infinite maturity, the price could be anywhere between S and S-K.

• What are the purpose and payoff of a European option with infinity maturity? If the option will never be exercised, why do you need this option? – Gordon May 27 '19 at 16:43
• Given that the option will never be exercised, it has a zero value . – Gordon May 27 '19 at 16:48
• @Gordon this is interesting because the math suggested differently .... hmm – Sanjay May 27 '19 at 18:55
• @Sanjay: the math is based on a maturity and then take a limit. However, if we start with a infinity maturity, then the payoff is not well defined, given that you will never have the chance to exercise -- this is mentioned in my first question on the payoff. – Gordon May 27 '19 at 20:04

So we are assuming that the stock does not pay any dividend? Then it should be $$S_0$$, which is the limiting behaviour of the BS formula. $$\Phi ( d_1)$$ goes to 1 as maturity approaches infinity, and $$\Phi (d_2)$$ can only be between 0 and 1, so the $$e^{-r\tau}$$ will make the $$K e^{-r\tau} \Phi (d_2)$$ zero.

I can see the argument around the payoff at infinity not of any value, but then if the stock never pays dividend, then can’t you make the same argument about the stock payoff? So call option price equal to $$S_0$$ make sense.

For a dividend paying stock, it should be zero as you can easily verify by looking at the extended BS formula. The second term goes to zero as above, but the first term now has got the term $$e^{-r_f \tau}$$ so it goes to zero as well. Again this makes sense because the stock value will mainly be coming from the dividend stream, which the option lacks, so its worth nothing.

I'm not sure there has ever existed a perpetual European call option, but I'm happy to indulge in the thought process. I say it can't be worth zero, because there are certain events that cause it to have value: a) if the option were subject to variation margin according to the market value, then obviously the market could decide the value is non zero. In this situation, theoretical arguments are moot- if you have sold the option, you will have to margin it, and if the other side of the trade has more liquidity than you, you will lose the battle. b) if there is no variation margin, you have a stronger argument- for one thing, who are you buying the option from? Given an infinite time frame, they will eventually go bankrupt. But there is still a potential for value: what if the underlying stock is subject to a corporate event such as a cash acquisition? Depending on the documentation, that could deliver cash to option holders. Even if the company goes bankrupt, there is sometimes residual value for stockholders.

So I say it cannot be zero, although you have a strong case that it is nowhere near the BS formula limit of S.

Value will be equal to the current price.You can solve by putting a limiting case on the BS formula. Solved : https://pasteboard.co/IgFAz95.jpg

For a dividend paying stock, the value will be always zero.

• check your formula for d1? I think this method should show C=S(0). – dm63 May 27 '19 at 21:14
• I think the interesting follow on question for those who believe the answer is zero, is: what is the value of a 1000 year call option. 100 years ? 10 years? Is there a maximum somewhere? It’s hard to believe that a 1000 year option is any different from an infinite option ! – dm63 May 27 '19 at 21:18
• Yeah, my calculations are a little wrong. The value will always be S0 for a non dividend paying stock. One thing we can clearly state using maths or common sense is that for a dividend paying stock holding an infinite maturity option is not beneficial at all bcoz we'll be missing out on cashflow, so it's value would definitely be zero. As for a non dividend paying stock, the value as suggested by the math will be equal to the stock price itself because an holding an infinite maturity option is as if I'm holding the stock perpetually! So at any instant the cost is S0 – Dhruv Mahajan May 27 '19 at 21:51
• That's my 2 cents. – Dhruv Mahajan May 27 '19 at 21:51

In the standard BS model the price follows a GBM so the company can never be worth zero. But, in practice, the underlying can become worthless.

In the GBM case,the price equals $$s_0$$. But, in reality there is a positive probability that $$S$$ becomes worthless and stays that way forever. Therefore, $$E[S_T]$$ must tend to zero as $$T$$ tends to infinity and the option must be worthless. If we find an asset that truly can keep its value forever, then $$s_0$$ seems fair? An example: at the end of times you have the option to jump back in time to today. Even though the point will never be reached, it will still always have the same value as your life has today.